# The Varchenko Determinant for Apartments

## TL;DR

This paper computes the Varchenko determinant for apartments in hyperplane arrangements, extending previous work on central arrangements and cones, by leveraging the structure of realizable conditional oriented matroids.

## Contribution

It introduces a method to compute the Varchenko determinant for apartments in hyperplane arrangements, generalizing prior results to a broader class of arrangements.

## Key findings

- Derived a formula for the Varchenko determinant for apartments.
- Extended the computation to realizable conditional oriented matroids.
- Connected the determinant computation to the structure of hyperplane arrangements.

## Abstract

Varchenko introduced a distance function on chambers of hyperplane arrangements that he called quantum bilinear form. That gave rise to a determinant indexed by chambers whose entry in position $(C,D)$ is the distance between $C$ and $D$: that is the Varchenko determinant. He showed that that determinant has a nice factorization. Later, Aguiar and Mahajan defined a generalization of the quantum bilinear form, and computed the Varchenko determinant given rise by that generalization for central hyperplane arrangements and their cones. This article takes inspiration from their proof strategy to compute the Varchenko determinant given rise by their distance function for apartment of hyperplane arrangements. Those latter are in fact realizable conditional oriented matroids.